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Question

A share of non-dividend paying stock is trading at USD 30. The maturity of both options is 1 year from now. A put with a strike of USD 28 is trading at USD 1 and call with a strike of USD 29 is trading at USD 8 The annual risk-free interest rate is 20%.

Is there an arbitrage opportunity? If so, demonstrate how an arbitrage profit can be calculated.

Answer

From my calculations I cannot find an arbitrage opportunity. I have tried various payoff tables,but still unsuccessful. I am preparing for my exams and am still finding these problems very confusing. Is there any tricks or hints anyone can give me.

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  • $\begingroup$ This is clearly too basic and thus out-of-scope. Hint: apply put/call parity to one of the two options. $\endgroup$ – LocalVolatility Feb 6 '17 at 18:33
  • $\begingroup$ Sadly its not basic for me :( $\endgroup$ – user2792941 Feb 6 '17 at 18:42
  • $\begingroup$ See quant.stackexchange.com/help/on-topic for what is in-/out-of-scope. My hint should help you figure this out. Otherwise, I suggest talking to your TA, classmates, ... $\endgroup$ – LocalVolatility Feb 6 '17 at 18:45
  • $\begingroup$ According to you hint both options are mispriced. So if I sell both of them still I cant find the answer. $\endgroup$ – user2792941 Feb 6 '17 at 19:39
  • $\begingroup$ That wasn't what I was hinting at. Think about one being expensive or cheap relative to the other. Again - I suggest you talk to your tutor or classmates. $\endgroup$ – LocalVolatility Feb 6 '17 at 20:39
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You could buy one share, sell one call, and buy one put. That would cost you \$23 (= 30 - 8 + 1).

A year later, if the stock were higher than \$29, the call buyer would call away the stock for \$29. You would net \$6. (Same is true if the stock were exactly \$29.)

A year later, if the stock were lower than \$28, you would exercise the put for \$28. You would net \$5. (Same is true if the stock were exactly \$28.)

A year later, if the stock were between \$28.01 and \$28.99, both options would expire worthless, and you could sell the stock for a net between \$5.01 and \$5.99.

Of course, you'd need to pay the piper. Your net would be reduced for borrowing the \$23 for a year. At 20%, this would be \$4.60.

You would be guaranteed \$0.40 (=\$5 - \$4.60).

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  • $\begingroup$ You're welcome. When LocalVolatility said "look for put/call parity," that was a clue to look at the call price vs. the put price. With everything close to the money (28 strike / 29 strike / 30 underlying), the put should be close in price to the call. But the put is $1 and the call is $8, and they are out of whack. At the heart of any arbitrage is selling the part that's too expensive / buying the part that's too cheap. $\endgroup$ – rajah9 Feb 7 '17 at 13:26

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